Random walks crossing power law boundaries

Studia Scientiarum Mathematicarum Hungarica

Volumn 34, Issue 1-3, 1998, Pages 219-252

  Kesten, H ^a   Maller, R A ^b  

^a Department of Obstetrics Gynecology   (United States)

^b UNIVERSITY OF WESTERN AUSTRALIA   (Australia)

Author keywords

Boundary crossing probabilities; Exit times; First passage times; Limsup behaviour; Random walks

Indexed keywords

EID: 0005459138     PISSN: 00816906     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (25)

1. CHOW, Y. S. and TEICHER, H., Probability theory. Independence, interchangeability, martingales, 2nd ed., Springer Texts in Statistics, Springer-Verlag, New York - Berlin, 1988. MR 89e:60001
2. CHOW, Y. S and ZHANG, C-H., A note on Feller's strong law of large numbers, Ann. Probab. 14 (1986), 1088-1094. MR 87j:60047
3. CHUNG, K. L., On the maximum partial sums of sequences of independent random variables, Trans. Amer. Math. Soc. 64 (1948), 205-233. MR 10, 132b
4. CHUNG, K. L. and FUCHS, W. H. J., On the distribution of values of sums of random variables, Mem. Amer. Math. Soc. (1951), no. 6, 12 pp. MR 12, 722e
5. ERICKSON, K. B., On the strong law of large numbers when the mean is undefined, Trans. Amer. Math. Soc. 185 (1973), 371-381. MR 49 #1579
6. d, Ann. Probab. 4 (1976), 802-828. MR 54 #14108
7. ESSEEN, C. G., On the concentration function of a sum of independent random variables, Z. Wahrsch. Verw. Gebiete 9 (1968), 290-308. MR 37 #6974
8. p-boundedness of the overshoot in multidimensional renewal theory, Ann. Probab. 23 (1995), 2022-2056. MR 97d:60119
9. GUT, A., Stopped random walks. Limit theorems and applications, Applied Probability. A Series of the Applied Probability Trust, 5, Springer-Verlag, New York - Berlin, 1988. MR 88m:60085
10. HAHN, M. G. and KLASS, M. J., Approximation of partial sums of arbitrary i.i.d. random variables and the precision of the usual exponential upper bound, Ann. Probab. 25 (1997), 1451-1470.
11. KESTEN, H., The limit points of a normalized random walk, Ann. Math. Statist. 41 (1970), 1173-1205. MR 42 #1222
12. KESTEN, H., A universal form of the Chung-type law of the iterated logarithm, Ann. Probab. 25 (1997), 1588-1620.
13. KESTEN, H. and MALLER, R. A., Infinite limits and infinite limit points of random walks, and trimmed sums. Ann. Probab. 22 (1994), 1473-1513. MR 95m:60055
14. KESTEN, H. and MALLER, R. A., Two renewal theorems for general random walks tending to infinity, Probab. Theory Related Fields 106 (1996), 1-38. MR 97i:60113
15. KESTEN, H. and MALLER, R. A., Divergence of a random walk through deterministic and random subsequences, J. Theoret. Probab. 10 (1997), 395-427.
16. KESTEN, H. and MALLER, R. A., Random walks crossing high level curved boundaries, J. Theoret. Probab. (1998).
17. n/n, Ann. Probab. 1 (1973), 719-757. MR 54 #14086
18. KLASS, M. J., Toward a universal law of the iterated logarithm. I, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 36 (1976), 165-178. MR 54 #3822
19. KLASS, M. J., Toward a universal law of the iterated logarithm. II, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 39 (1977), 151-165. MR 56 #6808
20. KLASS, M. J., The finite mean LIL bounds are sharp, Ann. Probab. 12 (1984), 907-911. MR 85j:60048
21. PRUITT, W. E., General one-sided laws of the iterated logarithm, Ann. Probab. 9 (1981), 1-48. MR 82k:60066
22. SIEGMUND, D., Sequential analysis. Tests and confidence intervals, Springer Series in Statistics, Springer-Verlag, New York - Berlin, 1985. MR 87h:62145
23. SPITZER, F., Principles of random walks, 2nd ed., Graduate Texts in Mathematics, Vol. 34, Springer-Verlag, New York, 1976. MR 52 #9383
24. STONE, C., The growth of a recurrent random walk, Ann. Math. Statist. 37 (1966), 1040-1041. MR 33 #5004
25. ZHANG, C-H., The lower limit of a normalized random walk, Ann. Probab. 14 (1986), 560-581. MR 87h:60131